Blessing and Curse: How a Supercapacitor’s Large Capacitance Causes its Slow Charging

The development of novel electrolytes and electrodes for supercapacitors is hindered by a gap of several orders of magnitude between experimentally measured and theoretically predicted charging time scales. Here, we propose an electrode model, containing many parallel stacked electrodes, that explains the slow charging dynamics of supercapacitors. At low applied potentials, the charging behavior of this model is described well by an equivalent circuit model. Conversely, at high potentials, charging dynamics slow down and evolve on two relaxation time scales: a generalized $RC$ time and a diffusion time, which, interestingly, become similar for porous electrodes. The charging behavior of the stack-electrode model presented here helps to understand the charging dynamics of porous electrodes and qualitatively agrees with experimental time scales measured with porous electrodes.

Figure 1

(a) Sketch of a supercapacitor containing a $1:1$ electrolyte, two porous electrodes, and a battery providing an electrostatic potential difference $2\mathrm{\Psi }$. (b) In our stack-electrode model, the cathode and anode each contain $n$ planar electrodes at intervals of $h$. Initial anionic and cationic densities are ${\rho }_b$ throughout the cell. At time $t=0$, $-\mathrm{\Psi }$ and $+\mathrm{\Psi }$ are applied to all electrodes on the left and right-hand sides of the system, respectively. (c) Equivalent circuit model for the stack-electrode model.

Figure 2

Time dependence of the scaled (a) potential $\varphi (x,t)$ and (b) the surface charge densities ${\sigma }_i(t)$ for $i=\{1,\dots ,5\}$ of the electrode for $\mathrm{\Phi }=0.001$, $\kappa L=100$, $H/L=1$, and $n=5$. The symbols and lines in (b) correspond to numerical and equivalent-circuit model calculations, respectively.

Figure 3

The surface charge density (a), (b) $\sigma (t)/{\sigma }_{eq}$, the salt concentration at the cell center (c), (d) $c(t)$, and the charge relaxation (e), (f) $1-\sigma (t)/{\sigma }_{eq}$ (solid lines) and the concentration decay $[c(t)-c_{eq}]/[c(0)-c_{eq}]$ (dashed lines) of the stack-electrode model for $\kappa L=100$, in (a), (c), and (e) for $n=1$, $H/L=0$ at potentials $\mathrm{\Phi }=\{0.01,0.1,1,2\}$, and in (b), (d), and (f) for $\mathrm{\Phi }=2$ and $H/L=1$, at $n=\{2,3,6\}$.

Figure 4

(a) The dependence of the (scaled) adsorption timescale ${\tau }_{\mathrm{ad}}$ on the (scaled) system size $(L+H{)}^2$ for $\kappa L=500$ at potential $\mathrm{\Phi }=2$ at several pore sizes $\kappa h=\{2,5,10,20,50\}$ and $n=\{2,5,10,20,50,100,150,200,300\}$, with $n=200$ and $n=300$ for $\kappa h=20$ and $\kappa h=50$ off the scale of the plot. (b) The prefactor $\alpha$ of [Eq. (5)] as a function of $\kappa h$ for $\kappa L=500$ with a linear fit through the data in (a); $\kappa L=300$ and 400 with identical results.